Method for modeling material constitutive behavior

ABSTRACT

In modeling material constitutive behavior, a methodology determines tool-chip friction and a position of a stagnation point on a cutting tool. The methodology includes measuring a ratio of cutting force to thrust force and measuring a chip thickness, h ch , produced by applying the cutting tool to a material. Initial values are estimated for tool chip friction and position of stagnation. The tool chip friction and position of stagnation are calculated to satisfy a specified relationship. Based on tool chip friction and position of stagnation, material strains, material strain-rates, material temperatures, and material stresses are calculated.

RELATED APPLICATIONS

This application claims priority to U.S. patent application Ser. No.60/574,090 filed on May 25, 2004, entitled “Method for Modeling MaterialConstitutive Behavior,” and is incorporated herein by reference.

TECHNICAL FIELD

The present invention relates to material modeling to determine strains,strain-rates, and temperatures.

BACKGROUND OF THE INVENTION

Modeling of material constitutive behavior in a variety of applicationsis important. The conventional tension or compression tests are onlyapplicable under low strain-rates (10⁻³-10⁻¹/s) and low temperatures.The accuracy of this method strongly depends on the models of chipformation and tool-chip friction. In the machining processes, chipdeformation, material constitutive relationships, and tool-chip frictionare coupled together and affect each other. In most of analytical modelsof chip formation, it is customary to calculate the strain-rate in theprimary shear zone by assuming the thickness ΔS of this zone to beone-tenth of the undeformed chip thickness. ΔS and strain-rates alsohighly depend on the tool edge roundness, i.e., the tool edge radius,the position of stagnation point on the rounded cutting edge, andtool-chip friction. The effect of tool edge roundness is neglected inthe parallel-sided shear zone model.

Thus, it would be an advancement in the art to provide a new slip-linemodel of chip formation for machining, taking into account the effectsof tool edge roundness on ΔS and strain-rates. Discussed herein is amethodology for modeling material constitutive behavior at largestrains, high strain-rates, and elevated temperatures through anorthogonal machining test.

SUMMARY

A system and methodology provides modeling of material constitutivebehavior at large strains, high strain-rates, and elevated temperaturesthrough an orthogonal machining test. The methodology is based on aslip-line model of chip formation proposed for machining with a roundededge tool. The model predicts strains, strain-rates and temperatures inthe primary shear zone. Tool-chip friction and the position ofstagnation point on the rounded cutting edge are determined by using areverse method, i.e., by matching the predicted and experimental forceratio and chip thickness. Extensive cutting tests involving 6061-T6,2024-T351, and 7075-T6 aluminum alloys over a wide range of cuttingconditions confirm the effectiveness of the proposed methodology.

Additional aspects will be apparent from the following detaileddescription of preferred embodiments, which proceeds with reference tothe accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

Various embodiments of the invention are now described with reference tothe figures, in which:

FIG. 1 is a cross-sectional view of a tool cutting a workpiece.

FIG. 2 is a cross-sectional diagram illustrating a slip-line model of atool and workpiece.

FIG. 3 is a hodograph associated with the cross-sectional diagram ofFIG. 2.

FIG. 4 is a cross-sectional view of a tool illustrating a slip-linemodel.

DETAILED DESCRIPTION

The presently preferred embodiments of the present invention will bebest understood by reference to the drawings, wherein like parts aredesignated by like numerals throughout. It will be readily understoodthat the components of the present invention, as generally described andillustrated in the figures herein, could be arranged and designed in awide variety of different configurations. Thus, the following moredetailed description of the embodiments of the apparatus, system, andmethod of the present invention, as represented in FIGS. 1 through 4, isnot intended to limit the scope of the invention, as claimed, but ismerely representative of presently preferred embodiments of theinvention.

Referring to FIG. 1, a cross-sectional diagram of a tool 10 engaging aworkpiece 12 in a cutting process. The workpiece 12 is cut at a depth tand a cutting speed v. Cutting speed refers to the speed at which thetool 10 moves with respect to the workpiece 12. Feed rate is the rate atwhich the workpiece 12 moves into the tool 10. Cutting speed and feedrate affect the time to finish a cut, tool life, finish of the machinedsurface, and power required of a cutting machine. Cutting speed may bedetermined by the material to be cut and the material of the tool. Feedrate may depend on several variables including width and depth of thecut and desired finish. The material layer 14 at the top is formed intoa chip 16 by a shearing process in a primary shear zone at AB. The chip16 slides up a rake face 18 and undergoes some plastic flow due tofrictional force.

FIG. 2 illustrates a slip-line model 200 for machining that includes atool 202 and a chip 204. The model 200 assumes that there is continuouschip formation with no built-up edge and straight chip formation. Thesize of the shear zones is exaggerated to clearly show slip-lines. Therounded cutting edge BN is approximately represented by two straightchords SB and SN. This follows a methodology that replaces a curvedfrictional boundary with one or multiple straight chords to simplify themathematical formulation of the slip-line problem. Point S is thestagnation point of material flow. Part of the material flows upwardfrom point S to B, while the other part flows downward from point S toN.

In FIG. 2, V_(c) is the cutting speed; V_(ch) is the chip speed; γ₁ isthe tool rake angle; δ is the angle between the straight boundary AA₂and V_(c); φ is the “shear-plane” angle; r_(n) is the tool edge radius;θ_(s) is the angle determining the position of the stagnation point S;and δ_(SB) and δ_(SN) are two slip-line angles. Each slip-linesub-region shown in FIG. 2 corresponds to a physical meaning. Thetriangular regions BHI, BB₃S, and SM₂N are caused by tool-chip frictionon the tool rake face 206, on the cutting edge SB, and on the cuttingedge SN, respectively. The region AA₁A₂ provides the material “pre-flow”effect. The primary shear zone corresponds to the parallel-sidedslip-line region that is directly associated with tool-chip frictionbelow the stagnation point S on the rounded cutting edge. The model 200includes the central fan region BB₁B₂ to enable the material to departfrom the rounded edge tool 202 in a direction tangential to the toolsurface.

FIG. 3 is a hodograph diagram 300 of the slip-line model 200. Velocitydiscontinuity is noted along slip-line AB₃S.

FIG. 4 is a model 400 illustrating a cutting tool 402. Four variablesare used to define a rounded cutting edge: (1) tool edge radius r_(n);(2) position θ_(s) of the stagnation point on the cutting edge; (3)tool-chip frictional shear stress τ_(SB) above the stagnation point onthe cutting edge; and (4) tool-chip frictional shear stress τ_(SN) belowthe stagnation point. In FIG. 4, τ_(rake) is the tool-chip frictionalshear stress on the tool rake face 404 and k is the average materialshear flow stress.

With reference to FIGS. 2 and 4, angles ζ_(SB), ζ_(SN), and ζ_(rake) aredetermined byζ_(SB)=[cos⁻¹(τ_(SB) /k)]/2  (1)ζ_(SN)=[cos⁻¹(τ_(SN) /k)]/2  (2)ζ_(rake)=[cos⁻¹(τ_(rake) /k)]/2  (3)

The chip thickness h_(ch) is calculated ash _(ch)=√{square root over (2)}·cos(π/4−ζ_(rake))·(BH·cos ζ_(rake)+SB·cos ζ_(SB))  (4)where,

$\begin{matrix}{{{SB} = {2 \cdot r_{n} \cdot {\sin( {\frac{\pi}{4} + \frac{\gamma_{1}}{2} - \frac{\theta_{s}}{2}} )}}};} & (5) \\{{{BH} = \frac{h_{c} + {{\sqrt{2} \cdot ( {{\Delta\; S} + {{{SB} \cdot \cos}\;\zeta_{SB}}} ) \cdot \sin}\;\delta} - {r_{n} \cdot ( {1 + {\sin\;\gamma_{1}}} )}}{( {{\cos\;\zeta_{rake}} + {\sin\;\zeta_{rake}}} ) \cdot {\sin( {\gamma_{1} + \zeta_{rake}} )}}};} & (6) \\{{\delta = {{\pi/4} - \gamma_{1} - \zeta_{rake}}};} & (7) \\{{\Delta\; S} = {\sqrt{2} \cdot r_{n} \cdot \sqrt{1 - \frac{\tau_{SN}}{k}} \cdot {{\sin( \frac{\theta_{s}}{2} )}.}}} & (8)\end{matrix}$

If the forces across A₂A₁I₁B₁, B₁B₂, B₂M₁, M₁M₂, M₂N are denoted by{right arrow over (F)}₁, {right arrow over (F)}₂, {right arrow over(F)}₃, {right arrow over (F)}₄, and {right arrow over (F)}₅, then theresultant force {right arrow over (F)} is:

$\begin{matrix}{\frac{\overset{->}{F}}{{kh}_{c}w} = {\frac{\overset{->}{F_{1}}}{{kh}_{c}w} + \frac{\overset{->}{F_{2}}}{{kh}_{c}w} + \frac{\overset{->}{F_{3}}}{{kh}_{c}w} + \frac{\overset{->}{F_{4}}}{{kh}_{c}w} + \frac{\overset{->}{F_{5}}}{{kh}_{c}w}}} & (9)\end{matrix}$where w is the width of cut. The dimensionless cutting forceF_(c)/kh_(c)w and thrust force F_(t)/kh_(c)w are determined bydecomposing F/kh_(c)w in directions parallel and normal to V_(c).

The material flow in the primary shear zone is far more clearly definedthan that along the tool-chip interface. Hence, the effort was focusedin the primary shear zone. The average shear strain γ in this zone iscalculated as:

$\begin{matrix}{\gamma = \frac{\cos\;\gamma_{1}}{{2 \cdot \sin}\;{\phi \cdot {\cos( {\phi - \gamma_{1}} )}}}} & (10)\end{matrix}$whereφ=γ₁+ζ_(rake)   (11)The total velocity discontinuity V_(s) for material to pass through theprimary shear zone is given by:

$\begin{matrix}{V_{s} = {\frac{\cos\;\gamma_{1}}{\cos( {\phi - \gamma_{1}} )} \cdot V_{c}}} & (12)\end{matrix}$The average shear strain-rate {dot over (γ)} in the primary shear zoneis then calculated as:{dot over (γ)}_(ave) =V _(s) /ΔS  (13)The average temperature T in the primary shear zone is given as:

$\begin{matrix}{T = {T_{W} + {\eta \cdot \lbrack {\frac{1 - \beta}{\rho_{s}S} \cdot \frac{{k \cdot \cos}\;\gamma_{1}}{\sin\;{\phi \cdot {\cos( {\phi - \gamma_{1}} )}}}} \rbrack}}} & (14)\end{matrix}$where T_(w) is the initial work temperature, η is a heattransfer-related constant, ρ_(s) is the material density, S is thespecific heat, and β is a parameter determined by the specific heat andthe thermal conductivity. Table 1 shows the inputs and outputs of amodel established above in Eqs. (1)-(14).

The goal is to determine a set of data combinations for k, γ, {dot over(γ)}, and T. The flow stress k can be known from F_(c)/kh_(c)wcalculated from Eq. (9) and F_(c) measured from a cutting test. In Table1, most of inputs including tool geometry, cutting conditions, and thephysical and thermal properties of work material can be known from givendata.

TABLE 1 INPUTS AND OUTPUTS OF THE MODEL ESTABLISHED AS EQS. (1)–(14).Inputs Tool Tool edge radius r_(n) geometry Tool rake angle γ₁ CuttingCutting speed V_(c) conditions Undeformed chip thickness h_(c) Width ofcut w Work Density ρ_(s) material Thermal physical and conductivity Kthermal Specific heat S properties Initial temperature T_(w) Tool-chipOn the tool rake friction face τ_(rake)/k On the tool edge SB τ_(SB)/kOn the tool edge SN τ_(SN)/k Position of θ_(s) stagnation point OutputsChip h_(ch), by Eq. (4) thickness Force ratio F_(c)/F_(t), by Eq. (9)Dimension- F_(c)/kh_(c)w and less forces F_(t)/kh_(c)w, by Eq. (9)Strain γ, by Eq. (10) Strain-rate {dot over (γ)}, by Eq. (13)Temperature T, by Eq. (14)

The unknown inputs are tool-chip friction (τ_(rake)/k, τ_(SB)/k, andτ_(SN)/k) and the position θ_(s) of the stagnation point. The remaininginputs r_(n), γ₁, V_(c), h_(c), w, ρ_(s), K, S, T _(w)are known inputs.V_(c), h_(c), and w are determined based on operator setup of thecutting tool. A reverse method is used to determine tool-chip frictionand θ_(s) by matching the predicted and experimental force ratio(F_(c)/F_(t)) and chip thickness (h_(ch)). Tool-chip friction is assumedto be uniform, i.e., τ_(rake)/k=τ_(SB)/k=τ_(SN)/k. In the reversemethod, the force ratio F_(c)/F_(t) and the chip thickness h_(ch) areexperimentally measured. Next, an initial estimate for tool-chipfriction and θ_(s) (i.e., τ_(rake)/k=0.5 and θ_(s)=10°) is made. Foraccuracy, θ_(s) should be less than 20°. As θ_(s) becomes equal to orgreater than 20° due to tool wear, the results will not be as accurate.A non-linear computation algorithm incorporating Eqs. (1)-(14) runs todetermine a specific combination of τ_(rake)/k and θ_(s) that satisfiesthe following condition:

$\begin{matrix}{{{\lbrack {( \frac{F_{c}}{F_{t}} )_{pr} - ( \frac{F_{c}}{F_{t}} )_{ex}} \rbrack^{2} + \lbrack {( \frac{h_{ch}}{h_{c}} )_{pr} - ( \frac{h_{ch}}{h_{c}} )_{ex}} \rbrack^{2}} \leq 10^{- 12}},} & (15)\end{matrix}$where the subscripts “pr” and “ex” stand for the predicted andexperimental values. Once tool-chip friction τ_(rake) and θ_(s) aredetermined, strains, strain-rates, temperatures, and stresses are allcalculated using the same Eqs. (1)-(14).

The number of machining tests needed to perform depends on themathematical form of material constitutive model to be adopted and onthe requirement for accuracy. In addition, it is recommended to includea wide range of cutting conditions in tests, so the established materialconstitutive model is more versatile.

A total of 108 orthogonal tube-cutting tests were conducted on a CNCturning center (HAAS SL-10) in the Machining Research Laboratory at UtahState University. The 108 cutting tests involved three materials(6061-T6, 2024-T351, and 7075-T6 aluminum alloys), three cutting speeds,and 12 feed rates. The material density is 2712.64, 2768.0, and 2795.68kg/m³, respectively. The data from Military Handbook (1998) was used toestablish the relationships between the specific heat S (JKg⁻¹K⁻¹), thethermal conductivity K (Wm⁻¹K⁻¹), and the temperature T (° C.) asfollows:

For AL-6061-T6:S(T)=4.96224×10⁻¹⁴ T ⁶−3.74891×10⁻¹¹ T ⁵−8.52635×10⁻⁹ T ⁴+1.48883×10⁻⁵ T³−5.08×10⁻³ T ²+1.17447T+856.20635  (16)K(T)=−1.60552×10⁻¹⁴ T ⁶+2.00589×10⁻¹¹ T ⁵−7.92838×10⁻⁹ T ⁴+7.77749×10⁻⁷T ³−1.41869×10⁻⁴ T ²+0.16685T+147.19982  (17)For AL-2024-T351:S(T)=−1.78389×10⁻¹³ T ⁶+1.6225×10⁻¹⁰ T ⁵−3.59302×10⁻⁸ T ⁴+3.25646×10⁻⁶ T³−2.69×10⁻³ T ²+1.12107T+862.38178  (18)K(T)=5.55104×10⁻¹⁵ T ⁶+5.77525×10⁻¹² T ⁵−9.30259×10⁻⁹ T ⁴+1.56712×10⁻⁶ T³+2.84875×10⁻⁴ T ²+0.187  (19)For AL-7075-T6:S(T)=−2.0×10⁻⁴ T ²+0.8764T+833.2  (20)K(T)=−3.0×10⁻⁴ T ²+0.1511T+128.18  (21)

The applicable ranges of temperatures for Eqs. (16-17) and Eqs. (18-21)are 0° C.-540° C. and 0° C.-450° C., respectively. Flat-faced toolinserts TPG432 KC8050 made of sintered carbide with a TiC/TiN/TiCNcoating on tool surface are employed. The tool insert has a working rakeangle of 5 degrees. The tool edge radius r_(n) is measured by using aMitutoyo type SV602 fine contour measuring instrument. The portion ofthe tool edge that had the most uniform distribution of r_(n) (58 μm) isemployed in the cutting tests. To avoid built-up edge and rapidtool-wear, the cutting speed varies at 200, 350, and 500 m/min. The feedrate changed from 63.8 to 223.3 μm/rev. The width of cut was 3.0 mm.

The cutting forces F_(c) and F_(t) are measured by employing a forcemeasurement system including a Kistler type 9257B piezo-electricthree-component dynamometer, a Kistler type 5814B1 multichannel chargeamplifier, and a computer data acquisition system (Labview). Themeasurement system frequency is much higher (far more than two times)than the frequency of the cutting forces. After the force signals arecollected from the dynamometer, MATLAB is employed to filter thehigh-frequency noise signal. The chip thickness h_(ch) is measured usingthree chip samples generated under the same cutting conditions. Theaverage value from these chip samples is taken as the experimentalh_(ch).

The von Mises material flow criterion is employed to relate the shearflow stress k, the shear strain γ, and the shear strain-rate {dot over(γ)} to the effective stress σ, the effective strain ε, and theeffective strain-rate {dot over (ε)} as follows:k=√{square root over (1/3)}·σ  (22)ε=√{square root over (1/3)}·γ  (23){dot over (ε)}=√{square root over (1/3)}·{dot over (γ)}  (24).The equation below shows the mathematical form of an alternative model:σ=[A+B·ε ^(n)][1+C·ln{dot over (ε)}*][1−T* ^(m)]  (25)where A, B, n, C, and m are five constitutive constants given in Table2; {dot over (ε)}* is the dimensionless strain-rate; and T* is thehomologous temperature.

TABLE 2 FIVE CONSTITUTIVE CONSTANTS Materials A B n C m 6061-T6 324 1140.42 0.002 1.34 2024- 265 426 0.34 0.015 1.00 T351 7075-T6 496 310 0.300.0 1.20

The five constants given in Table 2 are obtained from torsion, statictensile, dynamic Hopkinson bar tensile, or ballistic impact tests. Forexample, the material constants for AL-7075-T6 were obtained throughballistic impact tests with the impact velocities ranging from 270 to370 m/s and strains ranging from 0 to 1.0. These constants are validwithin the conditions under which their original tests were conducted.The constants for AL-6061-T6 and AL-7075-T6, given in Table 2, can stillbe employed for machining applications due to small prediction errors.However, relatively large prediction errors exist for AL-2024-T351.Hence, those constants for AL-2024-T351 listed in Table 2 need to bemodified for machining applications, if higher accuracy of predictionsis preferred or required. A set of modified constants for AL-2024-T351are given as: A=420 MPa, B=200 MPa, n=0.025, C=0.015, m=1.00. With thesemodified constants, the average difference D_(a) reduces to 7.3%, 6.9%,and 7.2% at the cutting speeds of 200, 350, and 500 m/min, respectively.

A new methodology for modeling material constitutive behavior through anorthogonal machining test is disclosed herein. The methodology has theadvantage of being directly applicable to the deformation conditions oflarge strains, high strain-rates, and high temperatures. The methodologycan also be employed to examine the applicability of existing materialconstitutive models. The methodology is efficient and convenient,compared to commonly employed techniques for testing materialstress-strain relationships. A new slip-line model takes into effect thetool edge roundness on strain-rates. A reverse methodology determinestool-chip friction and the position of a stagnation point.

The methodology applies to metal and metal alloys, which experienceplastic deformation in machining. The methodology also applies to metalmatrix composites materials that experience plastic deformation inmachining. Metal matrix composites are widely applied in modernindustries, such as aerospace, aircraft, automotive, sports, machinetool manufacturing, and the like. Examples of metal matrix materialsinclude aluminum matrix composites, magnesium matrix composites,titanium matrix composites, copper matrix composites, superalloymatrices composites, and metal-matrix composites in various laminatedforms. When applying the methodology to materials, tool wear must beavoided during cutting tests to ensure accuracy.

While specific embodiments and applications have been illustrated anddescribed, it is to be understood that the invention is not limited tothe precise configuration and components disclosed herein. Variousmodifications, changes, and variations apparent to those of skill in theart may be made in the arrangement, operation, and details of themethods and systems of the present invention disclosed herein withoutdeparting from the spirit and scope of the present invention.

1. A method for calculating a tool chip friction, τ_(rake), and aposition of stagnation, θ_(s), for a cutting tool, comprising: measuringa ratio of cutting force, F_(c), to thrust force, F_(t); measuring achip thickness, h_(ch), produced by applying the cutting tool to amaterial; estimating initial values for the tool chip friction,τ_(rake), and position of stagnation, θ_(s); calculating the tool chipfriction, τ_(rake), and the position of stagnation, θ_(s), to satisfy,${{\lbrack {( \frac{F_{c}}{F_{t}} )_{pr} - ( \frac{F_{c}}{F_{t}} )_{ex}} \rbrack^{2} + \lbrack {( \frac{h_{ch}}{h_{c}} )_{pr} - ( \frac{h_{ch}}{h_{c}} )_{ex}} \rbrack^{2}} \leq 10^{- 12}};$and applying the calculated tool chip friction, τ_(rake), and thecalculated position of stagnation, θ_(s), to a model of constitutivebehavior for the material.
 2. The method of claim 1, wherein calculatingthe tool chip friction, τ_(rake), and the position of stagnation, θ_(s),includes calculating h_(ch) from,$h_{ch} = {{\sqrt{2} \cdot \cos}\;{( {\frac{\pi}{4} - \zeta_{rake}} ) \cdot {( {{{{BH} \cdot \cos}\;\zeta_{rake}} + {{{SB} \cdot \cos}\;\zeta_{SB}}} ).}}}$3. The method of claim 2, wherein calculating h_(ch) includescalculating SB from,${SB} = {2 \cdot r_{n} \cdot {{\sin( {\frac{\pi}{4} + \frac{\gamma_{1}}{2} - \frac{\theta_{s}}{2}} )}.}}$4. The method of claim 2, wherein calculating h_(ch) includescalculating BH from,${BH} = {\frac{h_{c} + {{\sqrt{2} \cdot ( {{\Delta\; S} + {{{SB} \cdot \cos}\;\zeta_{SB}}} ) \cdot \sin}\;\delta} - {r_{n} \cdot ( {1 + {\sin\;\gamma_{1}}} )}}{( {{\cos\;\zeta_{rake}} + {\sin\;\zeta_{rake}}} ) \cdot {\sin( {\gamma_{1} + \zeta_{rake}} )}}.}$5. The method of claim 2, wherein calculating h_(ch) includescalculating δ from,δ=π/4−γ₁−ζ_(rake).
 6. The method of claim 2, wherein calculating h_(ch)includes calculating ΔS from,${\Delta\; S} = {\sqrt{2} \cdot r_{n} \cdot \sqrt{1 - \frac{\tau_{SN}}{k}} \cdot {{\sin( \frac{\theta_{s}}{2} )}.}}$7. The method of claim 1, wherein the material includes a metalcomposite material to experience plastic deformation in machining.
 8. Amethod for calculating a material shear strain, γ, of a material,comprising: determining a tool chip friction, τ_(rake), for a cuttingtool; determining a rake slip-line angle, ζ_(rake), based on the toolchip friction, τ_(rake); determining a shear plane angle, φ, based onthe rake slip-line angle, ζ_(rake), and a tool rake angle, γ₁;determining the shear strain, γ, based on the tool rake angle, γ₁, andshear plane angle, φ; and applying the shear strain, γ, to a materialconstitutive model.
 9. The method of claim 8, wherein determining a toolchip friction, τ_(rake), includes, measuring a ratio of cutting force,F_(c), to thrust force, F_(t); measuring a chip thickness, h_(ch),produced by applying the cutting tool to the material; estimatinginitial values for the tool chip friction, τ_(rake), and a position ofstagnation, θ_(s); and determining the tool chip friction, τ_(rake), andthe position of stagnation, θ_(s), that satisfies,${\lbrack {( \frac{F_{c}}{F_{t}} )_{pr} - ( \frac{F_{c}}{F_{t}} )_{ex}} \rbrack^{2} + \lbrack {( \frac{h_{ch}}{h_{c}} )_{pr} - ( \frac{h_{ch}}{h_{c}} )_{ex}} \rbrack^{2}} \leq {10^{- 12}.}$10. The method of claim 9, wherein determining the tool chip friction,τ_(rake), and the position of stagnation, θ_(s), includes calculatingh_(ch) from,h _(ch)=√{square root over (2)}·cos(π/4−ζ_(rake))·(BH·cosζ_(rake)+SB·cosζ_(SB)).
 11. The method of claim 10, wherein calculating h_(ch)includes calculating SB from,${SB} = {2 \cdot r_{n} \cdot {{\sin( {\frac{\pi}{4} + \frac{\gamma_{1}}{2} - \frac{\theta_{s}}{2}} )}.}}$12. The method of claim 10, wherein calculating h_(ch) includescalculating BH from,${BH} = {\frac{h_{c} + {{\sqrt{2} \cdot ( {{\Delta\; S} + {{{SB} \cdot \cos}\;\zeta_{SB}}} ) \cdot \sin}\;\delta} - {r_{n} \cdot ( {1 + {\sin\;\gamma_{1}}} )}}{( {{\cos\;\zeta_{rake}} + {\sin\;\zeta_{rake}}} ) \cdot {\sin( {\gamma_{1} + \zeta_{rake}} )}}.}$13. The method of claim 10, wherein calculating h_(ch) includescalculating δ from,δ=π/4−γ₁−ζ_(rake).
 14. The method of claim 10, wherein calculatingh_(ch) includes calculating ΔS from,${\Delta\; S} = {\sqrt{2} \cdot r_{n} \cdot \sqrt{1 - \frac{\tau_{SN}}{k}} \cdot {{\sin( \frac{\theta_{s}}{2} )}.}}$15. The method of claim 8, wherein determining a rake slip-line angle,ζ_(rake), includes calculating,ζ_(rake)=[cos⁻¹(τ_(rake) /k)]/2.
 16. The method of claim 8, whereindetermining a shear plane angle, φ, includes calculating,φ=ζ_(rake)+γ₁.
 17. The method of claim 8 wherein determining the shearstrain, γ, includes calculating,$\gamma = {\frac{\cos\;\gamma_{1}}{{2 \cdot \sin}\;{\phi \cdot {\cos( {\phi - \gamma_{1}} )}}}.}$18. The method of claim 8, wherein the material includes a metalcomposite material to experience plastic deformation in machining.
 19. Amethod for calculating an average strain-rate {dot over (γ)} in aprimary shear zone for a material by applying a cutting tool to thematerial, comprising: determining a tool chip friction, τ_(rake), and aposition of stagnation, θ_(s), for the cutting tool; determining a rakeslip-line angle, ζ_(rake), based on the tool chip friction, τ_(rake);determining a shear plane angle, φ, based on the rake slip-line angle,ζ_(rake), and a tool rake angle, γ₁; determining a total velocitydiscontinuity, V_(s), based on a tool rake angle, γ₁, a cutting speed,V_(c), and the shear plane angle, φ; determining a primary shear zonethickness ΔS based on θ_(s); determining an average strain-rate, {dotover (γ)}, based on the total velocity discontinuity V_(s), and theprimary shear zone thickness ΔS; and applying the average strain-rate,{dot over (γ)}, to a material constitutive model.
 20. The method ofclaim 19, wherein determining the tool chip friction, τ_(rake), and theposition of stagnation, θ_(s), includes, measuring a ratio of cuttingforce, F_(c), to thrust force, F_(t); measuring a chip thickness,h_(ch), produced by applying the cutting tool to the material;estimating initial values for the tool chip friction, τ_(rake), and theposition of stagnation, θ_(s); and determining a combination of the toolchip friction, τ_(rake), and the position of stagnation, θ_(s), thatsatisfies,${\lbrack {( \frac{F_{c}}{F_{t}} )_{pr} - ( \frac{F_{c}}{F_{t}} )_{ex}} \rbrack^{2} + \lbrack {( \frac{h_{ch}}{h_{c}} )_{pr} - ( \frac{h_{ch}}{h_{c}} )_{ex}} \rbrack^{2}} \leq {10^{- 12}.}$21. The method of claim 20, wherein determining the tool chip friction,τ_(rake), and the position of stagnation, θ_(s), includes calculatingh_(ch) from,$h_{ch} = {{\sqrt{2} \cdot \cos}\;{( {\frac{\pi}{4} - \zeta_{rake}} ) \cdot {( {{{{BH} \cdot \cos}\;\zeta_{rake}} + {{{SB} \cdot \cos}\;\zeta_{SB}}} ).}}}$22. The method of claim 21, wherein calculating h_(ch) includescalculating SB from,${SB} = {2 \cdot r_{n} \cdot {{\sin( {\frac{\pi}{4} + \frac{\gamma_{1}}{2} - \frac{\theta_{s}}{2}} )}.}}$23. The method of claim 21, wherein calculating h_(ch) includescalculating BH from,${BH} = {\frac{h_{c} + {{\sqrt{2} \cdot ( {{\Delta\; S} + {{{SB} \cdot \cos}\;\zeta_{SB}}} ) \cdot \sin}\;\delta} - {r_{n} \cdot ( {1 + {\sin\;\gamma_{1}}} )}}{( {{\cos\;\zeta_{rake}} + {\sin\;\zeta_{rake}}} ) \cdot {\sin( {\gamma_{1} + \zeta_{rake}} )}}.}$24. The method of claim 21, wherein calculating h_(ch) includescalculating δ from,δ=π/4−γ₁−ζ_(rake).
 25. The method of claim 21, wherein calculatingh_(ch) includes calculating ΔS from,${\Delta\; S} = {\sqrt{2} \cdot r_{n} \cdot \sqrt{1 - \frac{\tau_{SN}}{k}} \cdot {{\sin( \frac{\theta_{s}}{2} )}.}}$26. The method of claim 19, wherein determining a rake slip-line angle,ζ_(rake), includes calculating,ζ_(rake)=[cos⁻¹(τ_(rake) /k)]/2.
 27. The method of claim 19, whereindetermining a shear plane angle, φ, includes calculating,φ=ζ_(rake)+γ₁.
 28. The method of claim 19, wherein determining a totalvelocity discontinuity, V_(s), includes calculating,$V_{s} = {\frac{\cos\;\gamma_{1}}{\cos( {\phi - \gamma_{1}} )} \cdot {V_{c}.}}$29. The method of claim 19, wherein determining a primary shear zonethickness, ΔS, includes calculating,${\Delta\; S} = {\sqrt{2} \cdot r_{n} \cdot \sqrt{1 - \frac{\tau_{SN}}{k}} \cdot {{\sin( \frac{\theta_{s}}{2} )}.}}$30. The method of claim 19, wherein determining an average strain-rate,{dot over (γ)}, includes calculating,${\overset{.}{\gamma}}_{ave} = {\frac{V_{s}}{\Delta\; S}.}$
 31. Themethod of claim 19, wherein the material includes a metal compositematerial to experience plastic deformation in machining.
 32. A methodfor calculating an average temperature, T, of a material in a primaryshear zone when applying a cutting tool to the material, comprising:determining a tool chip friction, τ_(rake), and a position ofstagnation, θ_(s), for the cutting tool; determining a rake slip-lineangle, ζ_(rake), based on the tool chip friction, τ_(rake); determininga shear plane angle, φ, based on the rake slip-line angle, ζ_(rake), anda tool rake angle, γ₁; determining the average temperature, T, based oninitial temperature, T_(w), density ρ_(s), specific heat S, tool rakeangle, γ₁, shear plane angle, φ, and average material flow stress k; andapplying the average temperature, T, of the material in the primaryshear zone to a material constitutive model.
 33. The method of claim 32,wherein determining the tool chip friction, τ_(rake), and the positionof stagnation, θ_(s), includes, measuring a ratio of cutting force,F_(c), to thrust force, F_(t); measuring a chip thickness, h_(ch),produced by applying the cutting tool to the material; estimatinginitial values for the tool chip friction, τ_(rake), and the position ofstagnation, θ_(s); and determining a combination of the tool chipfriction, τ_(rake), and the position of stagnation, θ_(s), thatsatisfies,${\lbrack {( \frac{F_{c}}{F_{t}} )_{pr} - ( \frac{F_{c}}{F_{t}} )_{ex}} \rbrack^{2} + \lbrack {( \frac{h_{ch}}{h_{c}} )_{pr} - ( \frac{h_{ch}}{h_{c}} )_{ex}} \rbrack^{2}} \leq {10^{- 12}.}$34. The method of claim 33, wherein determining the tool chip friction,τ_(rake), and the position of stagnation, θ_(s), includes calculatingh_(ch) from,$h_{ch} = {{\sqrt{2} \cdot \cos}\;{( {\frac{\pi}{4} - \zeta_{rake}} ) \cdot {( {{{{BH} \cdot \cos}\;\zeta_{rake}} + {{{SB} \cdot \cos}\;\zeta_{SB}}} ).}}}$35. The method of claim 34, wherein calculating h_(ch) includescalculating SB from,${SB} = {2 \cdot r_{n} \cdot {{\sin( {\frac{\pi}{4} + \frac{\gamma_{1}}{2} - \frac{\theta_{s}}{2}} )}.}}$36. The method of claim 34, wherein calculating h_(ch) includescalculating BH from,${BH} = {\frac{h_{c} + {{\sqrt{2} \cdot ( {{\Delta\; S} + {{{SB} \cdot \cos}\;\zeta_{SB}}} ) \cdot \sin}\;\delta} - {r_{n} \cdot ( {1 + {\sin\;\gamma_{1}}} )}}{( {{\cos\;\zeta_{rake}} + {\sin\;\zeta_{rake}}} ) \cdot {\sin( {\gamma_{1} + \zeta_{rake}} )}}.}$37. The method of claim 34, wherein calculating h_(ch) includescalculating δ from,δ=π/4−γ₁−ζ_(rake).
 38. The method of claim 34, wherein calculatingh_(ch) includes calculating ΔS from,${\Delta\; S} = {\sqrt{2} \cdot r_{n} \cdot \sqrt{1 - \frac{\tau_{SN}}{k}} \cdot {{\sin( \frac{\theta_{s}}{2} )}.}}$39. The method of claim 32, wherein determining a rake slip-line angle,ζ_(rake), includes calculating,ζ_(rake)=[cos⁻¹(τ_(rake) /k)]/2.
 40. The method of claim 32, whereindetermining a shear plane angle, φ, includes calculating,φ=ζ_(rake)+γ₁.
 41. The method of claim 32, wherein determining theaverage temperature, T, includes calculating,$T = {T_{W} + {\eta \cdot {\lbrack {\frac{1 - \beta}{\rho_{s}S} \cdot \frac{{k \cdot \cos}\;\gamma_{1}}{\sin\;{\phi \cdot {\cos( {\phi - \gamma_{1}} )}}}} \rbrack.}}}$42. The method of claim 32, wherein the material includes a metalcomposite material to experience plastic deformation in machining.
 43. Amethod for calculating a chip thickness, h_(ch), produced by applying acutting tool to a material, comprising: determining a rake slip-lineangle, ζ_(rake), based on tool chip friction, τ_(rake), for the cuttingtool; determining an edge slip-line angle, ζ_(SB), based on the toolchip friction, τ_(rake); determining the chip thickness, h_(ch), basedon the rake slip-line angle, ζ_(rake), and the edge slip-line angle,ζ_(SB); and applying the calculated chip thickness, h_(ch) to a materialconstitutive model.
 44. The method of claim 43, wherein determining atool chip friction, τ_(rake), includes, measuring a ratio of cuttingforce, F_(c), to thrust force, F_(t); measuring a chip thickness,h_(ch), produced by applying the cutting tool to the material;estimating initial values for the tool chip friction, τ_(rake), and aposition of stagnation, θ_(s); and determining a tool chip friction,τ_(rake), and position of stagnation, θ_(s), that satisfies,${\lbrack {( \frac{F_{c}}{F_{t}} )_{pr} - ( \frac{F_{c}}{F_{t}} )_{ex}} \rbrack^{2} + \lbrack {( \frac{h_{ch}}{h_{c}} )_{pr} - ( \frac{h_{ch}}{h_{c}} )_{ex}} \rbrack^{2}} \leq {10^{12}.}$45. The method of claim 43, wherein determining the rake slip-lineangle, ζ_(rake), includes calculating,ζ_(rake)=[cos⁻¹(τ_(rake) /k)]/2.
 46. The method of claim 43, whereindetermining the edge slip-line angle, ζ_(SB), includes calculating,ζ_(SB)=[cos⁻¹(τ_(rake) /k)]/2.
 47. The method of claim 43, whereindetermining the chip thickness, h_(ch), includes calculating,$h_{ch} = {{\sqrt{2} \cdot \cos}\;{( {\frac{\pi}{4} - \zeta_{rake}} ) \cdot {( {{{{BH} \cdot \cos}\;\zeta_{rake}} + {{{SB} \cdot \cos}\;\zeta_{SB}}} ).}}}$48. The method of claim 43, wherein the material includes a metalcomposite material to experience plastic deformation in machining.
 49. Amethod for modeling material constitutive behavior, comprising:providing a cutting tool having a curved edge; applying the cutting toolto a material to remove one or more chips from the material; measuring aratio of cutting force, F_(c), to thrust force, F_(t), of the cuttingtool as it is applied to the material; measuring a chip thickness,h_(ch), of the one or more chips produced by applying the cutting toolto the material; calculating a tool chip friction, τ_(rake), and aposition of stagnation, θ_(s), based on the measured ratio of cuttingforce, F_(c), to thrust force, F_(t), of the cutting tool and the chipthickness, h_(ch); and providing the calculated tool chip friction,τ_(rake), and the position of stagnation, θ_(s) to a model of materialconstitutive behavior.